Question about the merge dimension of input channel and groups

[simpleshinobu]

Thank you for giving a fascinating tutorial in NeurIPS 2020 about equivariance, I am very interested and new in this field. After reading your paper, I have a question about the merging of the dimension of the input channel and groups. If we rotate the whole original image, in my understanding, the equivariant transformation is to permute the order in the group-dimension in the second feature representations (ignoring the translation). But if you merge this dimension into input-channels, how to keep the permutation equivariance in the following features? In short, for example, what is the operation for the last feature representations equivariant to the rotation for the whole original image.

[tscohen]

This is explained in the paper, eq. 11: https://arxiv.org/abs/1602.07576. In second and higher layers, we apply the same strategy as in the first, namely to rotate/transform the filter by each element in the group H, only now the meaning of "rotate/transform" is to apply a transformation to the spatial dimensions as well as a permutation of the orientation channels.